Series Convergence Tester

Does Σaₙ converge?

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About Series Convergence Tester

A series convergence tester applying multiple convergence tests. Tests include: ratio test, root test, comparison test, integral test, p-series test, and alternating series test. Shows which tests are conclusive and the result. Select from preset series. All calculations are client-side.

Series Convergence Tester Features

  • Ratio test
  • Root test
  • p-series
  • Alternating
  • Presets
Series convergence: does Σₙ aₙ have a finite sum? Tests: Ratio (L=lim|aₙ₊₁/aₙ|, L<1 converges), Root (L=lim|aₙ|^(1/n)), p-series (Σ1/nᵖ converges iff p>1), alternating (|aₙ| decreasing → 0).

How to Use

Select a series:

  • Σaₙ: Preset series
  • Tests: Applied automatically
  • Result: Converges/diverges

Test Hierarchy

  1. Divergence test first (aₙ→0?)
  2. Geometric/p-series recognition
  3. Ratio or root test
  4. Comparison/limit comparison
  5. Integral test (last resort)

Absolute vs Conditional

Absolute: Σ|aₙ| converges. Conditional: Σaₙ converges but Σ|aₙ| diverges. Alternating harmonic: conditional. Absolutely convergent series can be rearranged freely; conditionally convergent ones cannot (Riemann rearrangement theorem).

Step-by-Step Instructions

  1. 1Select a series.
  2. 2View tests applied.
  3. 3Check each result.
  4. 4Determine convergence.
  5. 5Identify test used.

Series Convergence Tester — Frequently Asked Questions

Which test should I try first?+

1) Divergence test (quick check). 2) Recognize geometric (ratio r) or p-series (1/nᵖ). 3) Ratio test (factorials, exponentials). 4) Root test (nth powers). 5) Comparison (similar to known series). 6) Integral test (continuous, decreasing function).

What if the ratio test gives L=1?+

Inconclusive! Try another test. Common examples: Σ1/n (diverges) and Σ1/n² (converges) both give L=1 for the ratio test. Use comparison, integral, or condensation test instead.

What is the Riemann rearrangement theorem?+

A conditionally convergent series can be rearranged to converge to ANY real number, or to diverge! Only absolutely convergent series are safe from rearrangement. This is why absolute convergence is 'stronger' and more useful.

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